Learning-by-Doing, Organizational Forgetting, and Industry Dynamics

Transcription

1 Learning-by-Doing, Organizational Forgetting, and Industry Dynamics David Besanko Ulrich Doraszelski y Yaroslav Kryukov z Mark Satterthwaite x September 4, 2004 Please do not distribute without permission of the authors Abstract It is generally believed that learning-by-doing acts a force that leads to market dominance. Because organizational forgetting can erode a learning-based cost advantage, it might be expected that organizational forgetting would make it less likely that learning-by-doing would lead to market dominance. In this paper we show that this traditional intuition may be incorrect. We analyze a fully dynamic model of price competition when rms face a learning curve and the possibility of organizational forgetting. We show that even though the leader rm may under-price the follower and this price di erence may grow as the leader s cost advantage widens, the market may remain unconcentrated in both the short run and long run. And even when learning-by-doing does give rise to a non-trivial degree of market concentration, a steepening of the learning curve does not necessarily translate into a higher degree of concentration. We also show that organizational forgetting does not act as a simple o set to the e ects of learning-by-doing. Rather, over an interesting range of parameters, organizational forgetting intensi es pricing rivalry and leads to a greater degree of market concentration. In other words, instead of serving as an antidote to market dominance, organizational forgetting makes a learning-based cost advantage more sustainable and thus makes it more likely that the market will be dominated by a single rm. By extending the model to include entry and exit, we show that predatory pricing can arise endogenously and that organizational forgetting makes predaotry behavior more likely to occur. We develop these insights by employing the framework in Ericson and Pakes (1995) to numerically analyze the Markov perfect equilibria (MPE) in a pricing game in a di erentiated products duopoly market. A striking feature of our analysis is that in contrast to recent papers that have employed this computational framework, we show that there can be multiple symmetric MPE. Kellogg School of Management, Northwestern University, Evanston, IL , U.S.A., d- besanko@kellogg.northwestern.edu. y Hoover Institution, Stanford University, Stanford, CA , U.S.A., doraszelski@hoover.stanford.edu. z Department of Economics, Northwestern University, Evanston, IL , U.S.A., kryukov@northwestern.edu. x Kellogg School of Management, Northwestern University, Evanston, IL , U.S.A., m- satterthwaite@kellogg.northwestern.edu. 1

2 1 Introduction Empirical studies provide ample support for the hypothesis that learning-by-doing can be an important source of cost advantage. Learning-based cost reductions have been documented in a wide variety of industrial settings including airframes (Alchian 1963), chemical processing (Lieberman 1984), construction of nuclear power plants (Zimmerman 1982), semiconductors (Irwin and Klenow 1994), and shipbuilding (Thompson 2001, Thornton and Thompson 2001). Empirical work has also begun to suggest the possibility that learningby-doing can be undone by organizational forgetting, i.e., the depreciation of the know-how that the rm has built up in the production process. Empirical evidence on organizational forgetting has been presented by Argote and Epple (1990) and Thompson (2003) for the case of World War II Liberty Ships, by Benkard (2000) in the production of wide-body airframes, and by Darr, Argote, and Epple (1995) in the operation of pizza franchises. It is generally believed that learning-by-doing acts a force that leads to market dominance. The idea is straightforward: a rm with a learning-based cost advantage can pro tably under-price its rivals for current sales, moving it further down the learning curve than its rivals. As the rm s cost advantage widens, it has even more leg room to under-price its rivals in the future, eventually allowing the rm to win nearly all sales. Since organizational forgetting can erode a learning-based cost advantage, one expects that organizational forgetting makes it less likely that learning-by-doing leads to market dominance. In other words, if learning-by-doing gives rise to a winner-take-all market, organizational forgetting acts as antidote to market dominance by making a rm s learning-based cost advantage less sustainable. In this paper we show that this traditional intuition may be incorrect. We analyze a fully dynamic model of price competition with di erentiated products when rms face a learning curve and the possibility of organizational forgetting. We show that even though the leader may under-price the follower a phenomenon Cabral and Riordan (1994) refer to as increasing dominance or ID and the price di erence may grow as the leader s cost advantage widens (increasing increasing dominance or IID), the market may remain unconcentrated in both the short run and long run. And even when learning-by-doing does give rise to a high degree of market concentration, a steepening of the learning curve does not necessarily translate into a higher degree of concentration. Contrary to the intuition above, we also show that organizational forgetting does not simply o set learning-by-doing. Rather, over an interesting range of parameters, organizational forgetting intensi es pricing rivalry and leads to a greater degree of market concentration. In other words, instead of serving as an antidote to market dominance, organizational forgetting can make a learning-based cost advantage even more sustainable and thus makes market dominance more likely. This result is particularly striking because in our model we employ a speci cation in which the likelihood of forgetting increases in the amount of know-how that the rm has learned, and so one suspects that a learning-based cost advantage would be especially fragile. 1 We develop these insights by employing the framework in Ericson and Pakes (1995) to numerically solve 1 As we discuss below, this speci cation is consistent with empirical evidence on learning and forgetting in industrial contexts. 2

3 the Markov perfect equilibria (MPE). A striking feature of our analysis is that in contrast to recent papers that have employed this computational framework, we show that there can be multiple symmetric MPE. These multiple equilibria are not an artifact of the mathematical properties of the demand and cost functions we use. Rather, they are grounded in the dynamics of the model and can be interpreted as a bootstrapping phenomenon spawned by organizational forgetting. For a given set of parameter values, di erent equilibria can give rise to very di erent industry structures in the short run and the long run. A given set of parameter values may give rise an equilibrium pricing function in which the leader charges a price that is lower, but not too much lower, than the follower s, allowing both rms to move down their learning curves in tandem. These same parameter values may give rise to a di erent equilibrium pricing function in which a rm that gains a cost advantage will charge extremely low prices in order to aggressively defend its advantage if it is in immediate danger of losing its cost advantage. Indeed, in some equilibria, a rm that has a signi cant knowhow lead will launch a price war to prevent the follower rm from moving into a position in which it might eventually pose a threat to the leader s cost advantage. In these equilibria, the resulting industry structure can be quite asymmetric. The distinctive role played by organizational forgetting is re ected in the fact that these multiple equilibria only arise in the presence of organizational forgetting. With no organizational forgetting, we nd a unique equilibrium in which each rm eventually reaches the bottom of its learning curve. A number of theoretical studies have explored the competitive implications of the learning curve, primarily focusing on quantity-setting games in nite-horizon models, including Spence (1981), Fudenberg and Tirole (1983), Ghemawat and Spence (1985), Ross (1986), and Cabral and Riordan (1997). The papers that are closest in spirit to ours are Cabral and Riordan (1994), Lewis and Yildirim (2002), and Benkard (2003). Cabral and Riordan (1994) analyze a symmetric MPE in a di erentiated-product duopoly market in which rms compete via prices to win a sale in each period. Cabral and Riordan focus on the circumstances under which equilibrium prices exhibit ID and IID. Our analysis di ers from Cabral and Riordan in that we consider a model with organizational forgetting and allow for the possibility that neither rm may make a sale in a particular period. Adding organizational forgetting makes it impossible to solve our model analytically, and so unlike Cabral and Riordan, we rely on a computational rather than an analytical approach to the characterization of the MPE. Because we compute the equilibrium pricing function, we can use the theory of Markov processes to determine the transient and ergodic distributions over states, which enables us to develop a characterization of industry dynamics that is much richer than would be possible through a purely analytical approach. A key insight from this analysis is, as noted above, that ID and IID are not su cient for an industry to be dominated by a market leader. 2 Lewis and Yildirim (2002) analyze a MPE between suppliers facing a learning curve, but they focus on the issue of how a single buyer (e.g., a government agency) can optimally design a multi-period procurement auction. This emphasis is very di erent from the objective of our paper which is to analyze competition 2 Nor, as we show, are ID and IID necessary for market dominance. 3

4 between rms facing a learning curve in an unregulated market in which buyers cannot a ect the dynamics of the industry. Like our paper, Benkard (2003) uses numerical methods to compute the MPE in a model of dynamic competition that includes learning-by-doing, organizational forgetting, and entry and exit. Benkard s objective is to calibrate his model to the wide-body jet market of the 1970s and 1980s, and he shows that the implied equilibrium dynamics closely track the observed dynamics of price competition in that industry. In contrast to Benkard s paper, our focus is not on calibrating our model to a particular industry setting. Rather, we show how underlying economic fundamentals can shape industry dynamics, with a particular focus on the role of organizational forgetting. The organization of the remainder of the paper is as follows. Section 2 describes the model and the approach used in our computations. Section 3 provides an overview of the equilibrium correspondence. Section 4 analyzes pricing behavior in equilibrium, and Section 5 characterizes the industry dynamics implied by it. Section 6 undertakes a number of robustness checks. Section 7 extends the model to include entry and exit and applies the insights of the model to analyze predatory pricing and limit pricing. Section 8 summarizes and concludes. 2 Model For expositional simplicity, we focus on an industry with two rms without entry and exit. The general model is outlined in Appendix B. 2.1 Firms and states We consider an in nite-horizon dynamic game in an industry that consists of two rms, indexed by n 2 f1; 2g. Firm n is described by its state e n 2 f1; : : : ; Mg, where a state describes a rm s stock of know-how. At any point in time, the industry is characterized by a vector of rms states e = (e 1 ; e 2 ) 2 2, where 2 is the state space. The notation e [2] denotes the state (e 2 ; e 1 ) found by switching the know-how levels of the two rms. The marginal cost c(e n ) of rm n depends on its accumulated know-how. The rms face a learning curve given by 8 < e n if 1 e n < m; c(e n ) = : m if m e n M; where = ln ln 2 = log 2 for a learning curve with a slope of percent. Thus, unit cost decreases by 1 percent whenever cumulative experience doubles. Coe cient is the marginal cost with minimal experience (normalized to be e = 1), and m < M represents the experience level at which the rm reaches the bottom of its experience curve. Following Cabral and Riordan (1994), we take a period to be a length of time that is just long enough for 4

5 at most one rm to make a sale. By making a sale, a rm can add to its stock of know-how. In contrast to Cabral and Riordan (1994), however, we also incorporate organizational forgetting in our model; see Argote and Epple (1990), Darr, Argote, and Epple (1995), Benkard (2000), and Thompson (2003) for empirical evidence. Accordingly, we assume that the evolution of the stock of know-how of rm n is governed the following law of motion: e 0 n = e n + eq n e fn ; where e n is the rm s know-how in the current period, e 0 n is its know-how in the next period, eq n 2 f0; 1g is the rm s quantity in the current period, and e f n 2 f0; 1g represents organizational forgetting. If eq n = 1, the rm augments its know-how through learning-by-doing, while if e f n = 1; the rm loses a unit of know-how through organizational forgetting. We let (e n ) = Pr( e f n = 1) denote the probability that rm n s know-how depreciates in the current period, and we assume that this probability is increasing in e n. Assuming that the probability of forgetting increases in the amount of know-how has several advantages. First, experimental evidence on forgetting in the industrial psychology literature suggests that the rate of forgetting is an increasing function of the amount learned (Bailey 1989). Second, by assuming that (e n ) is increasing, it can be shown that the expected decay of know-how in the absence of future learning is a convex function of time. 3 This phenomenon (known in the psychology literature as Jost s Second Law) is consistent with experimental evidence on forgetting by individuals (Wixted and Ebbesen 1991). Finally, a speci cation in which (e n ) increases in e n is conceptually similar to the capital-stock model employed in empirical work on organizational forgetting. In the capitalstock model, the depreciation of know-how is proportional to the existing stock of know-how, and so to counteract this, the accumulation of new knowledge through learning-by-doing must increase in proportion to the stock of existing know-how. 4 Our speci cation has the same property: as the rm s stock of know-how goes up, its probability of winning a sale must increase in order to counteract the depreciation of its knowhow. 5 In the computations described in the next section, we employ the functional form, (e n ) = 1 (1 ) en, where 2 [0; 1]. When > 0; this function is strictly increasing and concave in e n. If = 0; then (e n ) = 0 for all e n, and the rm never forgets. 6 Conditional on rm n making a sale in the current period (an event denoted by w), its know-how changes 3 A proof is available on request. If (e n) was constant in e n, the decay of know-how would be a linear function of time. 4 More speci cally, the law of motion in a capital stock model of organizational forgetting is e 0 n = (1 )e n + q n; where is the depreciation rate. 5 We do not employ a capital stock model of depreciation in this paper in order to keep the state space integer-valued. See Benkard (2003) for an alternative speci cation that approximates the capital stock model while maintaining an integer-valued state space. 6 One way to motivate this functional form would be to imagine that forgetting results when skilled workers leave the rm. In particular, suppose that each additional unit produced by the rm enables one more worker to acquire a special job-related skill that increases his/her productivity, and thus a unit e n of know-how corresponds to the number of workers in rm n who have acquired the special skill. If one or more of these specially skilled workers leaves the rm in the next period, the rm is assumed to lose, on net, the equivalent of one unit of experience. (Think of new workers being hired to replace those that leave, and assume that there is just enough organizational memory so that all but one of the new workers can be trained to replicate the skills of the workers they are replacing.) Under this interpretation, the forgetting parameter would represent the rate of labor turnover in a given period, and (e n) would be the probability that at least one worker from the group of e n workers who possess the special skill leaves the rm in the next period. 5

6 according the transition function 8 < Pr(e 0 1 (e n ) if e 0 n = e n + 1; nje n ; w) = : (e n ) if e 0 n = e n : Conditional on rm n not making a sale (an event denoted by l), its know-how changes according to the transition function 8 < Pr(e 0 1 (e n ) if e 0 n = e n ; nje n ; l) = : (e 0 n) if e 0 n = e n 1: At the upper and lower boundaries of the state space, we take the transition function to be Pr(MjM; w) = 1 and Pr(1j1; l) = 1, respectively. 2.2 Price competition The industry draws its customers from a large pool of potential buyers. In each period, one buyer enters the market and makes, at most, one purchase, either from one of the two rms in the industry ( inside goods 1 or 2) or an alternative product ( outside good 0) made from a substitute technology. 7 The net utility that a buyer obtains by purchasing product n is v n p n + e" n ; where v n is a deterministic component of net utility, and e" n is the buyer s idiosyncratic preference for product n. We assume that the deterministic utility of the inside goods is the same, so v 1 = v 2 = v: Further, we assume that the outside good is supplied under conditions of perfect competition that drive its price to its marginal cost c 0. A buyer s idiosyncratic preferences (e" 0 ;e" 1 ;e" 2 ) are unobservable to rms and are assumed to be iid Type 1 extreme value random variables with location parameter 0 and scale parameter. The scale parameter represents the extent of horizontal product di erentiation, with a lower value of corresponding to weaker product di erentiation. As! 0, the industry becomes a homogeneous product oligopoly. In any period, the buyer chooses the alternative that gives it the highest net utility. Given the assumed distribution of the idiosyncratic valuations, the probability that rm n makes a sale when prices are p (p 1 ; p 2 ) is given by the logit speci cation exp( v D n (p) = exp( v0 c0 ) + P 2 pn ) j=1 exp( v pj ): The demand function is symmetric (i.e., D 1 (p 1 ; p 2 ) = D 2 (p 2 ; p 1 )) and has three economically meaningful parameters:,v, and v 0 c 0. As v 0 c 0! 1 (i.e., the inside goods are in nitely better than the outside good), we are in the Cabral and Riordan (1994) setting in which the buyer always purchases from one of the two rms in the industry. 7 Since there is a di erent buyer in each period, buyers are non-strategic. Lewis and Yildirim (2002) consider a model of pricing along a learning curve when there is a single buyer who anticipates the dynamics of future competition. 6

7 2.3 Bellman equation Consider an industry that is in state e. Letting 2 (0; 1) denote a rm s discount factor, 8 the net present value of future cash ows to rm 1 is given by: V 1 (e) = max p 1 ( (p 1 c(e 1 ))D 1 (p 1 ; p 2 (e)) + where p 2 (e) denotes the price charged by the rival rm, and ) 2X D k (p 1 ; p 2 (e))v 1k (e) ; (1) k=0 V 1k (e) E[V 1 (e 0 )je; buyer purchases good k]; k 2 f0; 1; 2g; is the expectation of rm 1 s value function, conditional on the buyer purchasing good k in state e. 9 2 has a comparable value function. Let 1 (p 1 ) denote the term in brackets in equation (1). Di erentiating it with respect to p 1, and using properties of the logit demand speci cation 10, we 1 = D 1 1 (p 1 c(e 1 )) Di erentiating this again, combining terms, and using the expression in (2) gives us: Firm V : = 1 [1 2D 1 ] D 1 1 = 0 1 = 2 1 < 0, i.e., the objective function is strictly quasi-concave and the price choice p 1 (e) is therefore 1 equation: to 0 and rearranging terms, rm 1 s equilibrium price can be shown to solve the following p 1 = c(e 1 ) + 1 D 1 (p 1 ; p 2 (e)) (p 2 (e); e); (3) where (p 2 ; e) V 11 (e) (p2 )V 12 (e) + (1 (p 2 )) V 10 (e) 8 The discount factor is given by =, where r > 0 denotes the per period discount rate and 1 2 [0; 1) is an exogenous 1+r probability that the industry s order ow vanishes in the next period (e.g., because their products are supplanted by a drastic innovation that draws away its pool of customers). 9 To illustrate the form of V 1k (e); consider V 12 (e), the expectation of rm 1 s value function given that the buyer purchases from rm 2: V 12 (e) = Pr(e 1 1je 1 ; l) [Pr(e 2 je 2 ; w)v (e 1 1; e 2 ) + Pr(e 2 + 1je 2 ; w)v (e 1 1; e 2 + 1)] + Pr(e 1 je 1 ; l) [Pr(e 2 je 2 ; w)v (e 1 ; e 2 ) + Pr(e 2 + 1je 2 ; w)v (e 1 ; e 2 + 1)] : 10 In particular, we n = 1 Dn(p)(1 Dn(p)) n = 1 Dn(p)D k(p); k 6= n: 7

8 and exp( v (p 2 ) exp( v p2 p2 ) ) + exp( v0 c0 ) : The term (p 2 ; e) is rm 1 s (undiscounted) prize from making the next sale in state e when rm 2 charges price p 2 : It consists of the di erence between the rm s expected value V 11 (e) if it wins the sale and its expected value (p 2 )V 12 (e)+(1 (p 2 )) V 10 (e) if it does not win the next sale, where (p 2 ) is the probability that the rival wins the next sale, conditional on the rm not winning. The prize represents the wedge that makes dynamic pricing behavior di er from static pricing behavior. 2.4 Equilibrium Because the demand speci cation is symmetric, we focus attention on symmetric Markov perfect equilibria (MPE). Existence of a symmetric MPE in pure strategies follows from the arguments in Doraszelski and Satterthwaite (2004). In a symmetric MPE, it su ces to determine the value and policy functions of rm 1, and throughout the remainder of the paper, p (e) p 1 (e) denotes the symmetric equilibrium pricing function for rm 1, while V (e) = V 1 (e) denotes the corresponding equilibrium value function. 11 Further, we let V k(e) V nk(e); k = 0; 1; 2; denote the expected value functions. Finally, we let D (e) D 1 (p (e); p (e [2] )) denote the equilibrium probability that the typical rm makes a sale in state e. Given this notation, we can write the symmetric MPE as follows: h i V (e) = [p (e) c (e)] D (e) + (p (e [2] ))V 2(e) + 1 (p (e [2] ) V 0(e) : (4) p (e) = c (e) + 1 D (e) ; (5) where c (e) c(e 1 ) (p (e [2] ); e) is the virtual marginal cost. The virtual marginal cost is endogenously determined in equilibrium and equals the actual marginal cost minus the discounted prize from winning. equilibrium price p y (c) for marginal costs c = (c 1 ; c 2 ) is given by p y (c) = c D y (c) ; Since the (symmetric) static Nash where D y (c) D 1 (p y (c); p y (c [2] )), it follows that the MPE price p (e) is the static Nash equilibrium price corresponding to the virtual marginal costs c (e) = c (e); c (e [2] ) ; i.e., p (e) = p y (c (e)): 11 We thus obtain the value and pricing functions of rm 2 by switching the arguments of rm 1 s pricing and value functions: p 2 (e) = p 1 (e [2] ): V 2 (e) = V 1 (e [2] ): 8

9 2.5 Computation To compute a symmetric MPE, we adapt the algorithm described in Pakes and McGuire (1994). The algorithm works iteratively. It takes a value function V e (e) and pricing function ep (e) as the starting point for an iteration and generates updated value and policy functions. Each iteration proceeds as follows: 1. First, we use equation (5) to compute an updated pricing policy p (e) for rm 1, taking the pricing policies of rm 2 to be ep (e [2] ): In this step, we use V e (e) to compute the expected value functions, V 0(e); V 1(e); and V 2(e): 2. Next, we use equation (4) to compute an updated value function, V (e). 3. The iteration is completed by assigning V (e) to V e (e); and p (e) to ep (e): 12 The algorithm terminates once the relative change in the value and the policy functions from one iteration to the next is below a pre-speci ed tolerance. All programs are written in Matlab 6.5. Details are available from the authors upon request. 2.6 Parameterization In our numerical analysis, will focus on how the learning curve and organizational forgetting in uence the shape of the equilibrium pricing function and the industry dynamics that are implied by that pricing behavior. Accordingly, in our baseline analysis, we will report results for a wide range of values of and, holding xed all other parameters. Later, we will check whether the insights from our baseline analysis are robust to changes in the degree of product di erentiation, as measured by, and the size of the market of the inside goods, as parameterized by v 0 c 0 relative to v. The values of the parameters used in the computations are as follows Rate of learning We compute the equilibrium for the following values of : f0:95; 0:85; 0:75; 0:65; 0:55; 0:35; 0:15; 0:05g R Empirical estimates of experience curves tend to nd slopes in the range of 0:75 to 0:95 (Lieberman 1984, Dutton and Thomas 1984), and so in describing our results, we will often focus on the case = 0:85; which can be thought of (loosely) as the median slope across a wide array of empirical estimates. 12 More precisely, we assign a weighted average of V (e) and V ~ (e) to V ~ (e) to help the algorithm converge. 9

10 2.6.2 Rate of organizational forgetting We compute the equilibrium for the following values of : f0; 0:01; 0:02; 0:03; 0:04; 0:05; 0:06; 0:07; 0:08; 0:09; 0:10; 0:30; 0:50; 0:90g D Empirical studies of organizational forgetting that employ a capital stock model have found rates of forgetting that range from 4 percent per month to 25 percent per month. Our model is not strictly comparable to a capital stock model, so it is not immediately obvious what empirically plausible values of might be. To shed light on this question, Appendix A develops a mapping between the depreciation rate in a capital stock model and the parameter in our model. Based on this analysis, it would appear that the empirically relevant range of is between 0 and about Accordingly, in describing our results, we will focus our attention on cases in this range: = 0 (which will serve as a benchmark against which to compare our model to Cabral and Riordan (1994)), = 0:03, and = 0:08: As we will see below, this set of s generates a rich variety of industry dynamics Parameters held xed for the baseline analysis Attractiveness of the outside alternative, v 0 c 0 : In our baseline analysis, we set v 0 c 0 = 0. In Section 6, we discuss how the equilibrium is a ected if we make the outside alternative more attractive (v 0 c 0 = 3) and (in nitely) less attractive (v 0 c 0 = 1). Degree of horizontal di erentiation, : In our baseline analysis, we set = 1, resulting in a moderately weak degree of horizontal di erentiation. 13 In Section 6, we discuss how the equilibrium changes if horizontal di erentiation becomes stronger ( = 2) and extremely weak ( = 0:10) Parameters held xed throughout analysis State space; M: Throughout the analysis, we consider a state space with 30 possible levels of know-how, i.e., M = 30 Know-how level at which learning curve attens out, m: We assume that the learning curve attens out at 15 units of know-how, i.e., m = 15. Marginal cost at the top of the learning curve, : We consider a learning curve with an initial level of marginal cost equal to 10, i.e., = 10. Note that with = 10, along an 85 percent learning curve ( = 0:85), marginal cost falls from $10 per unit to approximately $5.30 per unit at e = To illustrate, if both rms set a price equal to their initial marginal cost = 10, the own price elasticity of demand would equal As rms drop their prices in tandem, the own-price elasticity of demand falls, reaching a level of if both rms set a price equal to the marginal cost at the bottom of an 85 percent learning curve. 10

11 Deterministic component of utility of the inside goods, v: Throughout the analysis, we x v = 10: Because v = 0, note that in the baseline case of v 0 c 0 = 0, when a rm is at the top of the learning curve, its product is on a par with the outside alternative. Discount factor, : We set = 0:9524: This corresponds to a variety of scenarios that di er according to the length of a period. For example, it corresponds to a case in which a period is one year, the industry is certain to survive, and the discount rate is 5 percent (r = 0:05). It could also correspond to a setting in which a period is one month, the market has a 96:5 percent chance of surviving from one month to the next, and the monthly discount rate is about 1.06 percent (which corresponds to an annual discount rate of about 12.7 percent) Equilibrium correspondence To characterize how the nature of learning-by-doing and organizational forgetting shapes equilibrium pricing behavior and industry dynamics, we compute the MPE for all (; ) combinations in P D, with all other parameters set to their baseline values. We then use the computed pricing function to construct the Markov transition probability of next period s state e 0 given the current state e. This allows us to use tools from stochastic process theory to analyze the Markov process of industry dynamics rather than relying on simulations. In particular, we can use the transition probabilities to compute the limiting distribution over states, 1 (e) implied by the equilibrium. Table 1 provides an overview of the MPE by showing the expected industry Her ndahl index corresponding to the limiting distribution: H 1 X e 1 " # X D (e) 2 + D (e [2] ) 2 D (e) + D (e [2] ) 2 1 (e): e 2 The Her ndahl index is based on the rms conditional market shares, and so it is bounded below by 0.5 and above by 1. Two points emerge from this overview. First, there may be multiple symmetric MPE for a given set of parameter values. Second, though the equilibria can di er greatly across parameter values, equilibrium pricing behavior can generally be classi ed into one of three categories that di er in interesting ways. We discuss each point in turn. 14 To put this second scenario in perspective, technology companies such as IBM, Cisco Systems, Microsoft, Intel, Dell, and Sun Microsystems had costs of capital in the range of 11 to 15 percent in the late 1990s. Further, an industry with a survival probability of 96.5 percent from one month to the next has an expected life of months. Thus, this second scenario might be thought of as broadly representative of a technology company that receives orders from customers on a once-a-month basis for a product in which the pace of innovative activity in the industry would be expected to make it obsolete in roughly two years. 11

12 0:95 0:85 0:75 0:65 0:55 0:35 0:15 0:05 F F T X F T X F T X F T X F T X F T X F T X 0 :500 :500 :500 :500 :500 :500 :500 :500 0:01 :500 :500 :500 :500 :500 :500 :500 :500 0:02 :500 :500 :500 :500 :500 :500 :500 :500 0:03 :500 :500 :516 :520 :521 :521 :500 :521 :500 :521 :500 :521 0:04 :502 :502 :513 :502 :519 :502 :521 :501 :521 :500 :523 :500 :523 :500 :523 0:05 :504 :530 :538 :580 :542 :527 :500 :527 :500 :527 0:06 :507 :559 :882 :923 :940 :500 :531 :500 :530 :500 :530 0:07 :511 :744 :884 :921 :938 :954 :500 :532 :500 :534 0:08 :515 :770 :883 :920 :937 :952 :501 :959 :501 :530 :961 0:09 :519 :787 :882 :918 :935 :950 :502 :957 :502 :960 0:1 :523 :803 :882 :916 :933 :949 :505 :956 :504 :958 0:3 :517 :860 :922 :936 :942 :945 :945 :945 0:5 :508 :597 :897 :956 :965 :969 :969 :968 0:7 :504 :539 :615 :762 :904 :965 :975 :977 0:9 :501 :511 :532 :561 :591 :672 :766 :803 Table 1: Expected Her ndahl index under limiting distribution for (; ) 2 R D. 12

13 Figure 1: Equilibrium correspondence (expected Her ndahl index) for = 0:85 and 2 D. 3.1 Multiple equilibria Table 1 illustrates that for some parameter combinations we were able to nd two, or even three, symmetric MPE. For example, for = 0:85; = 0:03; we found two equilibria: one in which the long-run expected Her ndahl is 0.500, and another in which the long-run expected Her ndahl is The multiple equilibria were discovered by means of a systematic search that employed various starting values for p () and V (): Speci cally, our approach is to start calculation from the equilibrium for a neighboring set of parameter values, and check if the solution is di erent from the already known equilibrium. Generically, we would expect that for any particular combination of parameter values, the number of symmetric MPE is odd (Herings and Peeters 2004). This suggests that for the case of = 0:85, the equilibrium correspondence has the shape shown in Figure 1, which in turn suggests that in this case there may be as many as three equilibria when = 0:03; of which we have computed two, those corresponding to points A or C on the equilibrium correspondence. Because our computational algorithm relies on iterated best responses, the equilibria that we have computed are stable in the sense that if we chose starting values close enough to the equilibria at A or C, the iterated best response algorithm will lead us back to the equilibria at A or C: By contrast, the conjectured equilibrium at B is likely to be unstable in the sense that even slight perturbations away from B will lead us either to A or C. In this respect, the equilibria we have calculated are appealing because one might imagine that a process of iterated best responses would be one 13

14 way that rms would work their way toward an equilibrium. Given the assumed properties of demand and cost, the static Nash equilibrium in our model can be shown to be unique. Further, we nd unique symmetric MPE in the absence of organizational forgetting. Thus, the multiple equilibria in our model seem to arise because of the dynamic structure of the model with organizational forgetting, and not because of any mathematical properties that ow from our speci cation of demand or the learning curve. Following our discussion of equilibrium pricing behavior and the resulting industry dynamics, we will develop an intuition for why the multiple equilibria arise in our model when there is organizational forgetting. 3.2 Classifying equilibria Table 1 suggests that there are three broad categories of equilibria. First, we have equilibria in which H 1 0:500 indicating that in the long run we end up with two equal-sized rms. An example is the case where = 0:85; = 0 (no organizational forgetting) or the rst equilibrium when = 0:85; = 0:03. As can be seen from the top two panels of Figure 2, except in a neighborhood of e = (1; 1), a rm s equilibrium price is not particularly sensitive to its own state or that of its rival. For this reason, we refer to these as at equilibria (labeled F in Table 1). Although a at equilibrium might exhibit a range of intense price competition in a neighborhood of e = (1; 1) what we will refer to as a well price competition in a at equilibrium is not particularly intense once rms begin to make modest progress in moving down their learning curves. Second, we have equilibria in which H 1 is between 0:500 to approximately 0:580; indicating that in the long run we end up with an asymmetric oligopoly in which one rm has a market share between 50 to 70 percent, while the other rm has a market share between 30 and 50 percent. An example is the second equilibrium when = 0:85; = 0:03: As can be seen from the graph of p (e) in the bottom left panel in Figure 2, the equilibrium pricing function not only has a well but it also exhibits a pronounced trench running along the diagonal. For this reason, we refer to these as trenchy equilibria (labeled T in Table 1). In a trenchy equilibrium, price competition between equally experienced rms is fairly intense. Price competition abates only when rms move into the asymmetric region of the state space. Finally, we have equilibria in which H 1 is extremely high (often in the range between about 0.7 and 0.95), indicating that in the long run we end up with a single rm that essentially dominates the market. An example of this is the case where = 0:85; = 0:08: As can be seen from the graph of p (e) in the bottom right panel of Figure 2, the equilibrium policy function not only has a deep trench running along the diagonal, it also has a sideways trench running along the edges of the state space. For this reason, we refer to equilibria of this type as extra trenchy (labeled at X in Table 1). In an extra-trenchy equilibrium, price competition between equally experienced rms is extremely intense; furthermore, there are regions in the state space where there is also intense price competition between rm with a signi cant cost advantage and its disadvantaged rival. 14

15 Figure 2: Price function for = 0:85 and = 0; 0.03 (both equilibria), Line in e 2 = 30 plane is the cost function c (e 1 ). Figure 3 shows the value functions corresponding to the equilibrium pricing functions in Figure 2. The value functions corresponding to the trenchy and extra-trenchy equilibria show that both a leader and a follower experience a drop in value as the industry moves from an asymmetric state near the diagonal to a state located along the diagonal; in other words, the diagonal trench on the pricing function is mirrored by a trench in the value function. Further, in the value function corresponding to the extra-trenchy equilibrium, the value of being the dominant rm is signi cant, while the value associated with being the follower is very close to zero. Further, the value associated with the initial state (1; 1) is also low, indicating that rms expect to dissipate signi cant pro tability in the race to become the industry leader. 15

16 Figure 3: Value functions for = 0:85 and = 0; 0.03 (both equilibria), For each (; ) pair in Table 1, we have classi ed the equilibria into one of three categories: F, T, X. Although we motivated the di erent classes of equilibria by pointing out that they give rise to signi cantly di erent expected Her ndahl indexes in the long run, the classi cation of equilibria in Table 1 is actually based on an algorithm that identi es wells, diagonal trenches, and sideways trenches through direct examination of the equilibrium pricing functions. An equilibrium is classi ed as at if the pricing function has no diagonal trench. An equilibrium is classi ed as trenchy equilibria if it has a diagonal trench, and either no sideways trenches or very narrow and shallow sideways trenches that are close to the axis of state space. An equilibrium is classi ed as extra-trenchy if it has a diagonal trench and su ciently deep sideways trenches that are several states wide. 16

17 4 Pricing behavior The three categories of equilibria give rise to di erent dynamic pricing behavior. In this section, we explore the nature of equilibrium pricing behavior in greater detail. 4.1 Learning-by-doing It is useful to compare the equilibrium pricing behavior in our dynamic model to the pricing behavior that would arise in the Nash equilibrium in a static model without learning or forgetting. We begin with a de nition: De nition 1 The value function V (e) is locally non-decreasing in its rst argument at state e if V (e 1 + 1; z) V (e 1 ; z) V (e 1 1; z) for z 2 fe 2 1; e 2 ; e 2 + 1g, and it is locally non-increasing in its second argument at state (e 1 ; e 2 ) if V (z; e 2 1) V (z; e 2 ) V (z; e 2 + 1) for z 2 fe 1 1; e 1 ; e 1 + 1g: The value function is well-behaved at e if it is locally non-decreasing in its rst argument and locally non-increasing in its second argument at (e 1 ; e 2 ): Our computations illustrate that the equilibrium value function is not guaranteed to be well behaved, although it typically is well behaved over much of the state space. When the value function is well behaved, we can unambiguously compare the MPE prices to the static Nash equilibrium prices: Proposition 1 If the value function V (e) is well behaved in state e, then a rm s prize from winning is non-negative, i.e., (p; e) 0. Furthermore, a rm s virtual marginal cost cannot exceed its actual marginal cost, and its equilibrium price in state e cannot exceed the static equilibrium price corresponding to the cost pair c(e) (c(e 1 ); c(e 2 )); i.e., c (e) c(e) and p (e) p y (c(e)): Proof 1 Note that h (p; e) (p) V 1(e) i h V 2(e) + (1 (p)) V 1(e) i V 0(e) : It is straightforward (but tedious) to establish that if V () is well behaved at e, V 1(e) V 2(e) 0 and V 1(e) V 0(e) 0, which implies (p; e) 0: This, in turn, implies that c (e) c(e 1 ) and c (e [2] ) c(e 2 ), so that c (e) c(e). Because the static Nash equilibrium price p y (c) is non-decreasing in c (Vives 1999), it follows that p (e) = p y (c (e)) p y (c(e)). This proposition highlights the fundamental economic impact of learning-by-doing. Whenever the value function is well-behaved, a rm anticipates a non-negative prize from winning the next sale. When this prize is strictly positive, the rm acts as if its marginal cost is lower than its current out-of-pocket cost c(e n ); inducing each rm, in equilibrium, to price more aggressively than it would have in a static Nash equilibrium. In e ect, the amount by which each rm lowers its price below the static Nash equilibrium price is a measure of the extent to which the rm uses price cuts to invest in lowering its future costs. 17

18 4.2 Organizational forgetting Because organizational forgetting counteracts the rate of learning and makes gains in know-how more transitory, one might expect that rms would be more reluctant to invest in the acquisition of know-how through aggressive price competition when there is organizational forgetting. But the diagrams in Figure 2 suggest a di erent conclusion: organizational forgetting can actually intensify price competition. When we have an 85 percent learning curve with no organizational forgetting ( = 0), we have a at equilibrium in which in nearly all states the equilibrium price is very close to the static Nash equilibrium price corresponding to the marginal cost c(m) at the bottom of the learning curve. This re ects the well-understood intuition that when the discount factor is close to 1, rms facing a deterministic learning curve price as if their marginal cost is c(m) (Spence 1981, Cabral and Riordan 1994). But if = 0:03, a modest degree of organizational forgetting, we obtain two equilibria, and in each of these equilibria there is fairly intense price competition at the top of the learning curve (in states near (1; 1)). Further, in the second of the two equilibria (in the lower right panel of Figure 2), there is also a diagonal trench where equally experienced rms engage in intense price competition. If organizational forgetting becomes even stronger ( = 0:08), price competition becomes even more intense throughout most of the state space. The intuitive explanation for the impact of organizational forgetting is this. Organizational forgetting creates a current that rms must ght against as they attempt to reduce costs by accumulating experience. When there is organizational forgetting, a rm increases its stock of know-how by making sales at a rate that exceeds the rate at which its know-how depreciates through organizational forgetting. When two rms have the same know-how, each rm will set the same price and have an equal probability of making a sale. This probability cannot exceed 0.5, and may be less than 0.5 depending on the attractiveness of the outside alternative. Given this, it may be virtually impossible for two equally experienced rms to move all the way down the learning curve, even though it might be quite possible for one rm to do so. This phenomenon gives rise to two opposing forces. On the one hand, organizational forgetting makes it less attractive for the rms to set low prices when they are at (or close to) the top of the learning curve: after all, why invest in accumulating know-how through price cuts when your know-how gains will be transitory (and when your rival will also be unable to sustain its know-how gains). This is the investment-sti ing e ect of organizational forgetting, and it works to soften price competition in states close to the top of the learning curve. On the other hand, though, because organizational forgetting can give rise to a situation in which the market can support just one low-cost rm, each rm has a strong incentive to move o and stay o the diagonal of the state space, even in states at the bottom of the learning curve. We call this the preemption e ect of organizational forgetting, and it works to intensify price competition. We can see the trade-o between the investment-sti ing and preemption e ects in the third and fourth columns of Table 2 which show the equilibrium prices p (1; 1) and p (2; 2) for symmetric rms at or near the top of an 85 percent learning curve. As increases from 0 to about , price competition at the top of the learning curve intensi es, indicating that the preemption e ect dominates the investment- 18

19 sti ing e ect over this range of. However, as increases beyond 0.08, price competition at the top of the learning curve begins to wane (although it still remains much more intense than it would be in the absence of organizational forgetting, until we reach the extremely high value of of 0.30). Thus, the investment-sti ing e ect dominates the preemption e ect only when becomes su ciently large. Equilib. type p (1; 1) p (2; 2) p (14; 14) p (15; 15) 0 F 7:80 7:78 7:24 7:24 0:01 F 7:64 7:65 7:22 7:23 0:02 F 7:10 7:37 7:19 7:22 0:03 F 4:87 6:25 7:14 7:21 0:03 T 2:89 4:03 5:99 6:13 0:04 F 1:25 3:17 6:80 6:99 0:04 T 1:36 3:03 5:65 5:94 0:05 T 0:83 1:31 4:84 5:06 0:06 T 3:63 1:71 4:41 4:56 0:07 X 5:08 3:95 4:02 4:13 0:08 X 4:59 4:01 3:05 3:21 0:09 X 3:94 3:84 2:10 2:25 0:10 X 3:19 3:52 1:25 1:37 0:30 X 8:84 5:60 5:54 5:55 0:50 X 10:80 8:82 5:20 5:52 0:70 X 11:00 9:24 5:37 5:71 0:90 X 11:20 9:37 5:57 5:90 Table 2: Prices in symmetric states at the top and the bottom of the learning curve, = 0:85. (Equilibrium classi cations: F = at equilibrium; T = trenchy equilibrium; X = extra-trenchy equilibrium.) The trade-o between the investment sti ing and preemption e ects is di erent if one or both rms happen to have reached the bottom, or close to the bottom, of the learning curve. At the bottom of the learning curve, the investment-sti ing e ect of organizational forgetting is virtually absent: cost reductions from accumulating further know-how are zero or close to zero anyway, so it really does not matter that much that organizational forgetting makes those small gains more transitory. However, the preemption e ect continues to operate. We see this in the fourth and fth columns of Table 2: when rms are at, or near the bottom, of an 85 percent learning curve (states (14; 14) and (15; 15)), increases in tend to intensify price competition. The preemption e ect may be so strong that a rm may be willing to set price below its marginal cost even when it is at the bottom of the learning curve. A straightforward extension of Theorem 4.4 in Cabral and Riordan (1994) tells us that this is something that would not happen without organizational forgetting: if = 0, the equilibrium price of a rm at the bottom of the learning exceeds its marginal cost no matter what the state of its rival rm, i.e., p (m; e 2 ) > c(m) for any e 2 2 f1; : : : ; Mg). 15 However, when there is organizational forgetting a rm that reaches the bottom of the learning curve might price below marginal 15 Cabral and Riordan prove their theorem in a model in which there is no outside demand alternative. The logic of their proof of Theorem 4.4 can be readily adapted to the setting in our model where there is an outside alternative. A proof is available on request. 19

20 cost. For example, when = 0:85, = 0:08, p (15; 14) = 4:07 and p (15; 15) = 3:21, both of which are less than the rm s marginal cost c(m) of 5:30: 4.3 Wells, trenches, and resultant forces The most dramatic manifestations of the intensi ed price competition brought about by organizational forgetting are the wells and trenches in the pricing function. Wells (as seen in the top right-hand panel and the two bottom panels of Figure 2) represent an intense preemption battle between the rms to establish an initial advantage, followed by a quick surrender by the follower once one of the rms has established an initial advantage. As this gure (as well as Table 2) shows, even a modest amount of organizational forgetting can give rise to intense races to acquire initial advantage. Diagonal trenches (as seen in the two bottom panels of Figure 2) correspond to price wars that are triggered when a follower moves from a position of cost disadvantage to one of cost parity. Diagonal trenches are fueled by the prize that accrues to a rm from assuming the role of industry leader and from avoiding the role of industry follower. Put another way, because organizational forgetting makes it more di cult for both rms to move down their learning curve in tandem, diagonal trenches re ect the strong incentive that each rm has to ght hard to move o and stay o the diagonal of the state space. Sideways trenches (as seen in the bottom right-hand panel of Figure 2) are, like diagonal trenches, price wars that are triggered when a follower improves its competitive position. In contrast to a diagonal trench, however, the price war in the sideways trench is not triggered because the follower has moved into a position of cost parity. Instead, it arises in order to keep a severely disadvantaged rm in its place as a marginal force in the industry. That is, a sideways trench is designed to prevent a rm with minimal know-how from breaking through to a position where it can evolve into a rm that could, at some point in the future, represent a threat to the pro tability of the market leader. If diagonal trenches are about ghting against an imminent threat, sideways trenches are about ghting a distantly imagined threat. One might think of them as an equilibrium manifestation of the dictum, Only the paranoid survive. Pricing behavior drives the evolution of the industry. The impact of wells and trenches on equilibrium behaviors can be highlighted through the use of resultant forces diagrams. Given that the industry is in state e in the current period, it will be in state e 0 in the next period, where e 0 k e k 2 f 1; 0; 1g; k = 1; 2. We determine the expected movement of the state by computing the probability weighted average of e 0 using the Markov transition probabilities. Reinterpreting e 0 e as a direction and the associated probability as the force operating in that direction, the expected movement E(e 0 eje) becomes the resultant force. 16 Figure 4 shows the resultant forces for the scenarios we have been focusing on: = 0:85 and = 0; 0:03, and 16 More formally, let (e 0 je; p (e)) denote the Markov transition probability from state e = (e 1 ; e 2 ) to state e 0 = (e 0 1 ; e0 2 ). We rst compute the resultant force as E[(e 0 e)je] = X (e 0 1 e 1 ; e 0 2 e 2 )(e 0 je; p (e)): e Then, we plot an arrow with the foot at (e 1 ; e 2 ) and the head at (e 1 ; e 2 ) + E[(e 0 e)je]. e 20